Decision Maths
Unit 6
Matchings (Bipartite Graphs)
As none of the girls
want to be attached to
Dave, a complete match
is not possible
Group X
Group Y
A
R
B
W
C
G
D
P
and if possible a complete match.
5
YOU MUST DISPLAY THIS FINAL MATCH
Initial Match E to 3
Initial Match E to 5
A
1
B
2
C
3
D
4
E
5
A
1
B
2
C
3
D
4
E
5
Complete Match
Complete Match
(A – 1) (B – 4) (C – 2) (D – 5) (E – 3)
(A – 1) (B – 4) (C – 3) (D – 2) (E – 5)
Group X
Group Y
Group Y
Group X
A
M
A
M
D
P
D
P
E
N
E
N
J
S
J
S
Original Bipartite Graph
Initial Match
Group X
Group X
Group Y
Group Y
A
M
A
M
D
P
D
P
E
N
E
N
J
S
J
S
Original Bipartite Graph
Complete match starting from E to P
E–P–D–N
(A,M) (D,N) (E,P) (J,S)
Group X
Group X
Group Y
Group Y
A
M
D
P
E
N
J
S
M
A
P
D
E
N
J
S
Original Bipartite Graph
Complete match starting from E to M
E–M–A– P–D–N
(A,P) (D,N) (E,M) (J,S)
Both of these solutions are equally valid
Adjacency matrix
A
B
C
D
E
F
X
1
2
3
4
5
6
0
0
0
0
0
1
0
0
0
1
1
0
0
1
0
0
0
1
1
0
1
0
1
0
1
1
0
0
0
1
0
1
0
1
0
0
Y
A
1
B
2
C
3
D
4
E
5
F
6
A–5┼ B -3┼ F –1
C–4┼ E -2┼ D –6
Final Match
(A ,5)
(B,3)
(C,4)
(D,6)
(E,2)
(F,1)
Matchings
By the end of the topic you should know how to
Draw a Bipartite graph from information given as
MATCHINGS or in an ADJACENCY TABLE
Draw an ADJACENCY TABLE from information
given in a bipartite graph.
From an INITIAL PAIRINGS draw a bipartite graph
that shows a complete match
Show the correct notation for each match, e.g. A – 3 ┼ E etc.
State the completed matched pairings as a FINAL MATCH
For example (A,3) (B,4) (C,1) (D,5) (E,2)
BIPARTITE GRAPHS
If you have to draw a bipartite graph you must show, if
asked for
The graph with all the matchings
The graph with the INITIAL matchings
Another graph to show the COMPLETE matchings,
with deletions (although this is not marked)
Show the notation for each matching (This is marked)
A – 3 ┼ C – 1 ┼ B etc.
State the completed matched pairings. (This is marked)
For example (A – 3) (B – 4) (C – 1) (D – 5) (E – 2)